P. Darwood P.N. Fletcher G.S. Hilton Indexing term: Antenna phased arrays, Beamforming, Pattern synthesis Abstract: Beam synthesis remains one of the most difficult problems for an antenna array designer, this task being made more difficult when one limits the number of elements to a handful. Pattern synthesis in planar arrays is often achieved by the sampling of a continuous aperture distribution at the element locations. In the case of arrays of limited numbers of elements this produces errors that can unacceptably distort the radiation pattern. When one also then considers the effects of mutual coupling between the antenna elements, the application of ideal element weights derived in this way are no longer valid. In the paper a technique is presented, using the concept of retrodirective beams, that can achieve a low sidelobe beam from a continuous distribution on a circular aperture. The theory is then extended to include a planar array of antenna elements. This is achieved by the addition of retrodirective beams to the uniform beam pattern of the array. Measured data is presented for a small planar array antenna using the derived method. A method for correction of the element weights, to minimise the effects of mutual coupling, producing low sidelobe radiation patterns is then applied. No knowledge of the antenna array’s coupling coefficients are required. 1 Introduction Efficient communications systems rely on the antenna to form the desired far-field radiation patterns (typically low sidelobe levels and narrow main beam). To make the cost of such systems more attractive, antenna arrays with small numbers of elements are 0 British Crown Copyright, 1998DERA Published with the permission of Her Britannic Majesty’s Stationery office IEE Proceedings online no. 19981989 Paper first received 27th October 1997 and in revised form 11th February 1998 P. Darwood and G.S. Hilton are with the University of Bristol, Centre for Communications Research, Queen’s Building, University Walk, Bris- tol BS8 lTR, UK P.N. Fletcher is with DERA, St. Andrews Rd., Malvem, Worcs., WR14 3PS, UK being sought. However, reducing the number of elements of an antenna array introduces a number of problems in pattern synthesis. Typically, the pattern synthesis of antenna arrays with circular boundaries is achieved by, first, calculating the continuous aperture distribution required to form the desired radiation pattern, and then sampling this at discrete points representing the locations of antenna array elements [l]. This is adequate when the number of antenna elements is large and thus the sampling of the aperture distribution retains most of the detail of the distribution. However, when the number of elements is small, the sampling of the aperture distribution can become inadequate, as the distributions required to achieve low sidelobes are often rapidly varying functions. This is analogous to the case of undersampling a signal. To restore the radiation pattern of the antenna array to that of the circular aperture, an iterative weight correction method can be applied [l]. Another method used to try and minimise the error introduced by sampling is known as integrated sampling [2], but, as noted in the paper, the accuracy of the method is not expected to be very good when applied to extremely small arrays. A second and more intractable problem is that of mutual coupling. When in an array, antenna elements experience an electromagnetic environment which can be very different, depending upon the elements location within the array. An element near the centre of the array is completely surrounded by other elements, whereas an element near the edge of the array is only partially surrounded. The presence of other antenna elements alters the current distribution on an antenna element, and thus the radiation pattern is corrupted. This means that, in a small antenna array, the elements can have very different radiation patterns, and, thus, applying ideal complex weights to achieve a desired beam can lead to degradation of the radiation pattern. Thus, a technique is required to apply perturbations to ideal element weights to minimise the effects of mutual coupling on the desired beam. In this paper, a pattern synthesis technique is developed using the concept of retrodirective beams to achieve low sidelobe radiation patterns in an ideal continuous circular aperture and an ideal planar array antenna. In the latter case, it is found that low sidelobe beams can be produced without the need for an iterative procedure as described in [ 11. However, when these ideal weights are applied to the measured radiation patterns of an experimental antenna array, IEE Proc.-Microw. Antennas Propag.. Vol. 145, No. 4, August 1998 344
Abstract: Beam synthesis remains one of the most difficult problems for an antenna array designer, this task being made more difficult when one limits the number of elements to a handful. Pattern synthesis in planar arrays is often achieved by the sampling of a continuous aperture distribution at the element locations. In the case of arrays of limited numbers of elements this produces errors that can unacceptably distort the radiation pattern. When one also then considers the effects of mutual coupling between the antenna elements, the application of ideal element weights derived in this way are no longer valid. In the paper a technique is presented, using the concept of retrodirective beams, that can achieve a low sidelobe beam from a continuous distribution on a circular aperture. The theory is then extended to include a planar array of antenna elements. This is achieved by the addition of retrodirective beams to the uniform beam pattern of the array. Measured data is presented for a small planar array antenna using the derived method. A method for correction of the element weights, to minimise the effects of mutual coupling, producing low sidelobe radiation patterns is then applied. No knowledge of the antenna array’s coupling coefficients are required.
1 Introduction
Efficient communications systems rely on the antenna to form the desired far-field radiation patterns (typically low sidelobe levels and narrow main beam). To make the cost of such systems more attractive, antenna arrays with small numbers of elements are 0 British Crown Copyright, 1998DERA Published with the permission of Her Britannic Majesty’s Stationery office IEE Proceedings online no. 19981989 Paper first received 27th October 1997 and in revised form 11th February 1998 P. Darwood and G.S. Hilton are with the University of Bristol, Centre for Communications Research, Queen’s Building, University Walk, Bris- tol BS8 lTR, UK P.N. Fletcher is with DERA, St. Andrews Rd., Malvem, Worcs., WR14 3PS, UK
being sought. However, reducing the number of elements of an antenna array introduces a number of problems in pattern synthesis. Typically, the pattern synthesis of antenna arrays with circular boundaries is achieved by, first, calculating the continuous aperture distribution required to form the desired radiation pattern, and then sampling this at discrete points representing the locations of antenna array elements [l]. This is adequate when the number of antenna elements is large and thus the sampling of the aperture distribution retains most of the detail of the distribution. However, when the number of elements is small, the sampling of the aperture distribution can become inadequate, as the distributions required to achieve low sidelobes are often rapidly varying functions. This is analogous to the case of undersampling a signal. To restore the radiation pattern of the antenna array to that of the circular aperture, an iterative weight correction method can be applied [l]. Another method used to try and minimise the error introduced by sampling is known as integrated sampling [2], but, as noted in the paper, the accuracy of the method is not expected to be very good when applied to extremely small arrays.
A second and more intractable problem is that of mutual coupling. When in an array, antenna elements experience an electromagnetic environment which can be very different, depending upon the elements location within the array. An element near the centre of the array is completely surrounded by other elements, whereas an element near the edge of the array is only partially surrounded. The presence of other antenna elements alters the current distribution on an antenna element, and thus the radiation pattern is corrupted. This means that, in a small antenna array, the elements can have very different radiation patterns, and, thus, applying ideal complex weights to achieve a desired beam can lead to degradation of the radiation pattern. Thus, a technique is required to apply perturbations to ideal element weights to minimise the effects of mutual coupling on the desired beam.
In this paper, a pattern synthesis technique is developed using the concept of retrodirective beams to achieve low sidelobe radiation patterns in an ideal continuous circular aperture and an ideal planar array antenna. In the latter case, it is found that low sidelobe beams can be produced without the need for an iterative procedure as described in [ 11. However, when these ideal weights are applied to the measured radiation patterns of an experimental antenna array,
thod requires no a priori array coupling coefficients
main lobe coincides a sidelobe of the main
two dimensions. circular bound- t for their anal-
ture. The ideal
-90 -60 -20
Fig. 1 Radiation pattern of a = 1.4h
The radiation pattern lar aperture scanned to
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U 30 60 90 onale off boresight, deg.
a mformly illumiizated circular apevature,
of a uniformly weighted circu- e,, 4, is
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where 6 is elevation azimuth angle 4 thr Fig. 1 (a = 1.4A). T
gives, for any cut at eam, the pattern of
to work, these side1 azimuth angles 4. A array, the far out sidelobes are relati s a retrodirective technique need only est sidelobes to the main beam. In th linear array two retrodirective beams , but in a circular aperture an infinite
F(Q, 4 )
= j r e - j k r sin(0,) cos(4,-o)ej /cr sin 0cos(4-o) rdrdg 0 0
(2) This will give a radiation pattern that has a peak at e,, & and thus can be used to cancel the sidelobe of the uniform beam at this point. However, the addition of a retrodirective beam designed entirely to cancel the sidelobe peak of a uniform beam does not afford con- trol of the rest of the resultant radiation pattern and could, consequently, affect the main beam in an unde- sired way. Thus, in choosing a retrodirective beam, fac- tors other than the direction of maximum radiation need to be taken into account. This can mean that the retrodirective beam peak (Os, #$) is not chosen to coin- cide exactly with the uniform beam sidelobe peak or that the phase and amplitude weighting (p) applied to the retrodirective beam is such that complete cancella- tion does not occur at the sidelobe peak. These points are illustrated in the following.
To cancel the sidelobes at all angles of 4, i.e. the ring sidelobe, scanned beams such as eqn. 2 are required for all angles of 4,. Combining such scanned beams, for all angles of &, results in a retrodirective beam that is able to cancel the whole of the ring sidelobe of the uniform beam. This retrodirective beam is given by
2n a 2n
e jk r s in 0 cos(4-0) rdrdad4,
rdrdo e j k r sin 0 cos(+-o)
a
0
x (Jo(krs in0))rdr (3)
The first Bessel function in eqn. 3 is the aperture distri- bution. The retrodirective beam is chosen to have a null on boresight in order not to affect the main beam of the uniform pattern, thus the elevation scan angle for the retrodirective beam Os, is chosen to be the angle off boresight of the first null of the uniform radiation pattern (eqn. 1). For a circular aperture of radius a = 1.4it, this is calculated to be 6, = 25.9". Substituting this value into eqn. 3 gives a retrodirective beam with a null on boresight as desired (Fig. 2)
s The magnitude and phase of the retrodirective beam (complex scalar p in eqn. 4) are chosen such that, when combined with the uniformly weighted beam, the ring sidelobe is cancelled. It is not always required to
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cancel the ring sidelobe at its peak (i.e. the peak of the retrodirective pattern is not at the same position as the sidelobe of the uniform pattern) and thus ,U can be arbitrarily chosen, resulting in the sidelobe being can- celled at an angle e,. For 0, = 34.4", the resultant far- field radiation pattern has 30dB sidelobes, Fig. 2.
-LO 1 \[ ,
90 angle o f f boresight,deg.
Retrodirective and low sidelobe beams for a circular aperture, a = Fig.2 1.4h - low sidelobe .......... retrodirective
Fi .3 si&obe and Taylor continuous aperture distributions, a = 1.4h __ low sidelobe aperture .......... Taylor aperture
Comparison of sidelobe level against beamwidth for the new low
Fig. 3 demonstrates how the half-power beamwidth for the resultant pattern compares with that of a Taylor distribution on the circular aperture. The Taylor distribution performs marginally better with respect to this criterion. However, when the angle 0, is chosen to be the maximum value of the sidelobe of the uniform beam (approximately 36"), the resultant beam performs as well as the Taylor distribution, producing a radiation pattern with approximately -3 1 dB sidelobe levels.
3 Ideal planar antenna array
The preceding analysis has concentrated on an ideal continuous circular aperture for which low sidelobe dis- tributions are well established, i.e. Taylor distribution. Traditionally, to produce low sidelobes from a small ideal planar array, a continuous aperture distribution is sampled in the array-element locations. This is, how- ever, often inaccurate, due to the sampling error intro- duced by a small number of elements. This error can be sometimes overcome by 'correcting' the sampled weights using an iterative procedure [I]. For a planar array antenna the uniformly weighted beam is given by
Euntform ~ ~ ( 0 , 4 ) e ~ k r n sinQcos(4-cm) (0, $) I n
( 5 ) where An(@, @) is the element pattern and r,, on, are the position co-ordinates of the rzth element. In the absence of mutual coupling, all A,(@ @) are identical and are the 'element factor', this can then be removed from the sum in eqn. 5 leaving the array factor. To scan the above beam to a position e,, @s an additional phase weighting is applied to eqn. 5. The scanned beam is given by eqn. 6:
E(Q, 4) = A,(0, $)e--3krn sin(0s)c"s(6a-un)
n
(6) e j k r , sin 6' c o ~ ( 4 - u ~ )
This beam can be considered as a retrodirective beam formed to cancel the ring sidelobe at one position e,, 4, when added to the uniformly weighted array. How- ever, to cancel the whole of the ring sidelobe, retrodi- rective beams such as eqn. 6 are required for each angle q&. Combining all of these retrodirective beams together, as in the case for the circular aperture, results in a combined retrodirective beam that should be able to cancel the whole of the ring sidelobe of the uniform beam:
d4s x e - j k r , sin(0,) eos(4,-cn)
Rearranging eqn. 7 reveals the weights required to pro- duce the combined retrodirective beam:
E ~ ~ ~ ~ ~ ( Q , ~ ) = A,(Q, 4)eJkrn sinecos(4-cn) n
27r
0 s
( 8 )
1 1 e - j k r n sin(Q,) C O S ( ~ ~ - - O ~ )
27r 0
This can then be simplified to
E+~(B, 4 ) = ~ A , ( Q , 4)eJkr , sin6cos(4-u~) n
x Jo(-kr, sin(Q,)) (9) where the Bessel term is the element weighting. As pre- viously stated for the circular aperture, the value of 6, is chosen so that the retrodirective beam has a null on boresight. Once again, combining the uniform and ret- rodirective beams in differing ratios varies the sidelobe levels of the new beam.
This method was applied to an experimental commu- nications planar array, consisting of 19 printed dipoles above a groundplane. The array lattice was hexagonal, with an inter-element separation of 0.56A. Initially, the array was considered to be ideal without mutual cou- pling (i.e. using the element factor). Varying sidelobe levels up to approximately 4 3 dB were obtained, com- pared with the uniform sidelobe level of -15 dB. In comparison with the iterative weight correction method [ 13, the retrodirective beam method is less computation- ally intensive as it does not require an iterative correc- tion procedure.
formed with the element 3 (eqn. 9). Combining these as would produce 30dB
minimise the error tern by mutual cou the radiation patter
methods for improving with ideal weights have
Coniparison of the radiation patterns for the experimental planar array
Directivity, dBi Average sidelobe level, dB Peak sidelobe level, dB
(Fig. 4) 18.0 -25.7 -12.6
16.2 -26.7 -1 5.2
(Fig. 5) 17.2 -27.6 -16.0
(Fig. 6) 16.5 -31.9 -20.9
experimental planar array
ideal antenna arrays, i.e. in coupling. It was shown in [Sl
(b) A technique where the individual active element weights are altered to minimise the error between the radiation pattern and that of the ideal array.
V
Fig. 4 Uniform radiation pa
(a) By combining beams in a different (Section 3).
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0.5
0
-0.5
- 1
contour levels,dB
!tern
:he uniform and retrodirective ratio than for the ideal array case
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-1 -0.5 0 0.5 1 I 1
contour levels,dB
been tried: Fig.5 Radiation pattern using method (a) I 1
0.5
v c
-0.5
- 1
- 1 -0.5 0 0.5 1
Method (a): In order for the sidelobes of the uni- form beam to be cancelled, the retrodirective beam has to be phased and weighted appropriately. This is achieved by noting where the ring sidelobe of the uni- form beam is ( I3 Z 42") and then calculating the aver- age sidelobe magnitude and phase on the I3 = 42" contour. This is repeated for the same contour of the retrodirective beam. The retrodirective beam is weighted such that the I3 = 42" contour has the same average magnitude and the average phase is in antiphase with that of the uniform beam. These beams are then combined. The resultant radiation pattern pro- duced in this way is shown in Fig. 5 , and it can be seen to be an improvement on the uniform beam. However, from Table 1 it is seen that the radiation pattern is only marginally improved upon that obtained with the application of ideal weights.
Method ( 6 ) : Applying the ideal weights (wn) calcu- lated in eqn. 9 to the active element patterns of the 19 element planar array (Active,(B, $)) does not give the desired radiation pattern (Ed@ $))
&(e, 4) = W n I S O n ( @ , 4 ) # w,Active,(o, 4) n n
(10) where Iso,(I3, $) is the ideal array element factor. This is due to the effects of mutual coupling. To minimise the error between these two patterns, we allow pertur- bations dw, to the active element weights. Thus we
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desire the best solution 6w, to eqn. 11:
n n
By rewriting eqn. 11 using w’, = w, + 6w,, we require a solution i d n to
n n
where w’, are the new active element weights. This can be performed using Galerkin’s method, where the active element patterns are the basis and testing func- tions [7]. The resulting matrix equation can be easily solved for w’,.
Calculating the new weights w’,, for the active-ele- ment patterns from ideal weights w,, results in the radi- ation pattern shown in Fig. 6. As can be seen from the Figure and from Table 1 the radiation pattern is much improved, with a reduction of more than 5dB in the peak and average sidelobe levels over the beam for the case when ideal weights are applied.
V
1
0.5
0
.0.5
- 1
- 1 - 0.5 0 0.5 1 U
contour levels, dB
Fig. 6 Radiation pattern using method (b)
This approach to far-field pattern synthesis of a small planar array has successfully resulted in a pencil beam with low sidelobe radiation characteristics. Fur- thermore, the presented technique requires no knowl- edge of the antenna array coupling coefficients as described in [3, 41. The proposed method is not neces-
sarily optimal in any sense, but serves to demonstrate the applicability of the retrodirective beam concept to the synthesis of planar arrays.
5 Conclusions
The theory of retrodirective beams in connection with planar array pattern synthesis has been explored. It has been shown how the application of a retrodirective beam can reduce the sidelobe levels of a continuous cir- cular aperture. This goal has been achieved previously, for instance the Taylor circular aperture distribution. The method is then extended to include discrete arrays of antenna elements. In the ideal case of array pattern synthesis (i.e. no mutual coupling), it is shown that the retrodirective beam is a useful tool for sidelobe level reduction. In particular, there is no need for iteration of the element weights in order to achieve the desired sidelobe levels, as is the case when a continuous aper- ture distribution is sampled at a small discrete number of element locations.
The method is extended to include actual measured active element patterns and it is found that, due to the corruption of the active element patterns by mutual coupling, the ideal weights are only partially successful in reducing the sidelobes of the radiation pattern. Two methods to improve on this situation are then dis- cussed, with the second (Galerkin’s method) yielding much improved results. A principal benefit of this tech- nique is that the coupling coefficients of the antenna array are not required.
6 Acknowledgments
The authors wish to thank Prof. J. P. McGeehan for the provision of facilities in the Centre for Communica- tions Research. This work was supported by DERA, Malvern.
References
ELLIOTT, R.S.: ‘Antenna theory and design’ (Prentice-Hall 1981), Chap. 6 HODGES, R.E., and RAHMAT-SAMII, Y . : ‘On sampling con- tinuous aperture distributions for discrete planar arrays’, ZEEE Trans., 1996, AP-44, (11), pp. 1499-1508 STEYSKAL, H., and HERD, J.S.: ‘Mutual coupling compensa- tion in small array antennas’, ZEEE Trans., 1990, AP-38, (12), pp.
DARWOOD, P., FLETCHER, P.N., and HILTON, G.S.: Mutual coupling compensation in small planar array antennas’,
IEE Proc., Microw. Antennas Propug., 1998, 145, (l), pp. 1-6 FLETCHER, P.N., and DEAN, M.: ‘Application of retrodirec- tive beams to achieve low sidelobe levels in small phased arrays’, Electron. Lett., 1996, 32, (6), pp. 506-508 BALANIS, CA.: ‘Antenna theory analysis and design’ (John Wiley & Sons, 1982), Chap. 14 HARRINGTON, R.F.: ‘Matrix methods for field problems’ in HANSEN, R.C. (Ed.): ‘Moment methods in antennas and scat- tering’ (Artech House, 1990), pp. 44-57
